evaluating garch models

8/3/2019 Evaluating GARCH Models

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Journal of Econometrics 110 (2002) 417435

www.elsevier.com/locate/econbase

Evaluating GARCH models

Stefan Lundbergh, Timo Terasvirta

Department of Economic Statistics, Stockholm School of Economics, P.O. Box 6501,

SE-113 83 Stockholm, Sweden

Abstract

In this paper, a unied framework for testing the adequacy of an estimated GARCH model is

presented. Parametric Lagrange multiplier (LM) or LM type tests of no ARCH in standardized

errors, linearity, and parameter constancy are proposed. The asymptotic null distributions of the

tests are standard, which makes application easy. Versions of the tests that are robust against

nonnormal errors are provided. The nite sample properties of the test statistics are investigated

by simulation. The robust tests prove superior to the nonrobust ones when the errors are non-

normal. They also compare favourably in terms of power with misspecication tests previously

proposed in the literature.c 2002 Elsevier Science B.V. All rights reserved.

JEL classication: C22; C52

Keywords: Conditional heteroskedasticity; Model misspecication test; Nonlinear time series; Parameter

constancy; Smooth transition GARCH

1. Introduction

When modelling the conditional mean, at least when it is a linear function of para-

meters, the estimated model is regularly subjected to a battery of misspecication tests

to check its adequacy. The hypothesis of no (conditional) heteroskedasticity, no error

autocorrelation, linearity, and parameter constancy, to name a few, are tested using

various methods. As for models of conditional variance, such as the popular GARCH

model, testing the adequacy of the estimated model has been much less common in

practice. But then, misspecication tests do exist for GARCH models as well. For ex-

ample, Bollerslev (1986) already suggested a score or Lagrange multiplier (LM) test

for testing a GARCH model against a higher order GARCH model. Li and Mak (1994)

Corresponding author.

E-mail address: [email protected] (T. Terasvirta).

0304-4076/02/$ - see front matter c 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 3 0 4 - 4 0 7 6 ( 0 2 ) 0 0 0 9 6 - 9


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418 S. Lundbergh, T. Terasvirta / Journal of Econometrics 110 (2002) 417 435

derived a portmanteau type test for testing the adequacy of a GARCH model. Engle and

Ng (1993) considered testing the GARCH specication against asymmetry using the

so-called sign-bias and size-bias tests. Chu (1995) derived a test of parameter constancy

against a single structural break. This test has a nonstandard asymptotic null distribu-tion, but Chu provided tables for critical values. Recently, Lin and Yang (1999) derived

another test against a single structural break, based on empirical distribution functions.

In this article, we present a number of simple misspecication tests for GARCH

models. The idea is to make misspecication testing easy without sacricing power.

We consider testing the null of no ARCH in the standardized errors, linearity or sym-

metry against a smooth transition GARCH, and parameter constancy against smoothly

changing parameters. A single structural break is nested in the alternative hypothesis

of our parameter constancy test. A two-regime asymmetric GARCH model such as

the so-called GJR model (Glosten et al., 1993) is nested in the alternative of smooth

transition GARCH. Note that the LM test of Bollerslev (1986) ts into our framework.

Furthermore, we show that under normality the portmanteau test of Li and Mak (1994)

is asymptotically equivalent to our test of no remaining ARCH. All our tests are LM

or LM type tests and require only standard asymptotic distribution theory. They may

be obtained from the same root by merely changing the denitions of the elements

of the score vector corresponding to the alternative hypothesis.

Very often in applications, the assumption of a normal error distribution of a GARCH

process is too restrictive. A useful feature of our tests is that robustifying them against

nonnormal errors is straightforward. In fact, our Monte Carlo experiments suggest that

one should always apply the robust versions of our tests. At the sample size 1000, usedin our simulation study and rather typical for GARCH applications, the eciency loss

compared to the nonrobust tests appears to be minimal when the errors are normal.

On the other hand, the same simulations show that the nonrobust tests tend to be

undersized if the error distribution is leptokurtic and that, in this situation, robust

tests have clearly better power than the nonrobust ones. In terms of power, the tests

proposed in this paper compare favourably with corresponding tests currently available

in the literature. This is true for all three cases we consider: no remaining ARCH,

linearity, and parameter constancy.

The plan of the paper is as follows. In Section 2, we dene the model. In Section

3, we discuss testing the null of no ARCH in the standardized errors and compare ourLM-test with the portmanteau test of Li and Mak (1994). Section 4 considers testing

null hypotheses of linearity and parameter constancy. Section 5 contains results of a

simulation experiment and Section 6 concludes.

2. The model

Consider a time series model with the following structure:

yt = f(wt;D) + t; (2.1)where f is at least twice continuously dierentiable with respect to D , for all wt=(yt1; u

t) with yt1 = (1; yt1; : : : ; ytn)

Rn+1 and exogenous ut= (u1t; : : : ; ukt) Rk,


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S. Lundbergh, T. Terasvirta / Journal of Econometrics 110 (2002) 417 435 419

everywhere in . The error is parameterized as

t = th1=2t ; (2.2)

where {t} is a sequence of independent identically distributed random variables withmean zero, unit variance and E3t = 0. Furthermore, the conditional variance ht = W

stsuch that st = (1;

2t1; : : : ;

2tq; ht1; : : : ; htp)

and W = (0; 1; : : : ; q; 1; : : : ; p) with

0 0, whereas 1; : : : ; q; 1; : : : ; p satisfy the conditions in Nelson and Cao (1992)

that ensure the positivity of ht. These conditions allow some of the parameters to be

negative, unless p = q = 1. Model (2.2) is thus the standard GARCH(p; q) model.

We assume regularity conditions hold such that the central limit theorem and the

law of large numbers apply whenever required. For such conditions in the multi-

variate GARCH(p; q) case, see Comte and Lieberman (2000). In the univariate case,

their conditions require the density of t

to be absolutely continuous with respect

to the Lebesgue measure and positive in a neighbourhood of the origin. Further-

more, it is required that E8t , which of course implies further restrictions on thedensity of t.

The assumption E3t = 0 that Comte and Lieberman (2000) do not need guarantees

block diagonality of the information matrix of the log-likelihood function. However,

it is not just a technical simplication. We shall consider, among other things, a test

that has power against asymmetric response to shocks. In deriving such a test it is

appropriate to assume that the conditional distribution of t given ht, is not skewed.

3. Testing the adequacy of the GARCH model

3.1. Testing the null hypothesis of no remaining ARCH

In order to consider the adequacy of the GARCH model, we formulate a parametric

alternative to the model. Assume that in (2.2),

t = ztg1=2t ; (3.1)

where {zt} is a sequence of independent, identically distributed random variableswith zero mean, unit variance and Ez

3t = 0. Furthermore, gt = 1 + ?

vt, where vt =(2t1; : : : ;

2tm)

and ? = (1; : : : ; m); j 0; j = 1; : : : ; m. Eq. (3.1) may thus be

written as

t = zt(htgt)1=2 (3.2)

and could be called an ARCH nested in GARCH model as 2tj = 2tj=htj ; j =

1; : : : ; m. We want to test H0: ? = 0 against ?= 0 and thus follow the standard prac-tice of choosing a two-sided alternative although the elements of ? are constrained

to be non-negative. Under this hypothesis, gt 1, and the model collapses into aGARCH(p; q) model. (For ways of testing H0: ? = 0 against ? 0 when ht

0, see

Lee and King (1993) and Demos and Sentana (1998).)We introduce the following notation. Let t and ht be the error t and the con-

ditional variance ht, respectively, estimated under H0, and 2

t = 2t=ht. Furthermore,


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let xt = h1

t 9ht=9W (9ht=9W

denotes 9ht=9W estimated under H0) and vt = (

2

t1; : : : ;

2

tm). The quasi maximum likelihood approach leads to the following result:

Theorem 3.1. Consider the model (3.2) where gt = 1 + ?vt and {zt} is a sequence

of independent identically distributed random variables with zero mean; unit variance

and Ez3t = 0. Under H0: ? = 0; the statistic

LM = (1=4T)

T

t=1

(2t=ht 1)vt

V(W)1

T

t=1

(2t=ht 1)vt

; (3.3)

where

V(W)1 = (4T=k)

Tt=1

vtv

t T

t=1

vtx

t

T

t=1

xtx

t

1 Tt=1

xtv

t

1

with k=(1=T)T

t=1(2t=ht1)2 is a consistent estimator of the inverse of the covariance

matrix of the partial score under the null hypothesis; has an asymptotic 2 distribution

with m degrees of freedom.

Proof. See the appendix.

The test may also be carried out using an articial regression as follows.

1. Estimate the parameters of the conditional variance model under the null, compute

2t=ht 1; t= 1; : : : ; T , and SSR0 =

Tt=1(

2t=ht 1)2.

2. Regress (2t=ht 1) on xt; vt and compute the sum of squared residuals, SSR1.

3. Compute the test statistic LM = T(SSR0 SSR1)=SSR0 or the F-version:

F =(SSR0 SSR1)=m

SSR1=(T

p

q

1

m)

:

For the sample sizes relevant in GARCH modelling, there is no essential dierence

between the properties of LM and F.

If the standard GARCH parameterization is assumed to be the true data-generating

process, then it is well-known in practice that the iid error process often cannot be

assumed normal, although the likelihood is constructed under the assumption of nor-

mality. It is therefore desirable to make the test robust against nonnormal errors. The

robust version of the LM test can be constructed following Wooldridge (1991, Proce-

dure 4.1). The test is carried out as follows:

1. Obtain the quasi maximum likelihood estimate for W under H0, compute 2t=ht1; vtand xt; t= 1; : : : ; T .

2. Regress vt on xt, and compute the (m 1) residual vectors rt; t= 1; : : : ; T .


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3. Regress 1 on (2t=ht 1)rt and compute the residual sum of squares SSR from that

regression. The test statistic

LMR = T SSRhas an asymptotic 2 distribution with m degrees-of-freedom under the null

hypothesis.

The purpose of the second step is to purge the eects of the normality assumption

from vt. The resulting test statistic has the same asymptotic distribution as (3.3) and is,

under normality, asymptotically ecient; see Wooldridge (1991). Any other consistent

estimator of W may be employed in place of the quasi-maximum likelihood one.

3.2. A portmanteau test and a comparison

Li and Mak (1994) recently introduced a portmanteau statistic for testing the ade-

quacy of the standard GARCH(p; q) model. The null hypothesis is that the squared and

standardized error process is not autocorrelated. Let r = (r1; : : : ; r m) be the m1 vector

of the rst m autocorrelations of {2t=ht} so that H0:r = 0. Assuming normal errors,Li and Mak (1994) showed that under this hypothesis

Tr is asymptotically normally

distributed where T is the number of observations and r = (1=2T)

(2t=ht 1)vt witht= ytf(wt; D); ht = ht(wt; D; W) and vt = (2t1=ht1 1; : : : ; 2tm=htm 1). Further-more, under the null hypothesis the asymptotic covariance matrix of Tr has the form

Vr(W) = Im Xr(W)G1(W)Xr(W) (3.4)

as plimTT1

vt v

t = 2Im under H0, and Xr(W) = (1=2T)

xtv

t . In (3.4),

G1(W) is a consistent estimator of the relevant block of the information matrix, eval-

uated at W = W. The portmanteau statistic becomes

Q(m) = TrVr(W)

1r (3.5)

which, as Li and Mak (1994) showed, is asymptotically 2-distributed with m degrees

of freedom under the null hypothesis. In order to link (3.5) to our theory in the case

of normal errors, dene

Q(m) =1

4T

(

2t=ht 1)v

t

Vr (W)

1

vt (2t=ht 1)

; (3.6)

where Vr (W) = ( 1=2T)(

vt v

t

vt x

t(

xtx

t)1

xtv

t ). The only dierence be-

tween (3.5) and (3.6) is the choice of the consistent estimator of the covariance matrix.

Vr (W) contains the matrix T1

vt v

t whereas its expectation 2Im appears in Vr(W).

The two covariance matrices are thus asymptotically equal so that the statistics (3.5)

and (3.6) and thus (3.3) and (3.5) have the same asymptotic null distribution. Thereason for introducing (3.3) as Li and Mak (1994) already derived the same test is

that obtaining the test using the LM principle makes it easy to see how the test can be


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made robust against nonnormal errors. Furthermore, the present test is can be viewed

as a natural extension of Engles (1982) classic LM test of no ARCH to this situation.

It is well-known that the McLeod and Li (1983) portmanteau test and the Engle (1982)

test are asymptotically equivalent, see, for example, Luukkonen et al. (1988b). Whenht = 0 in (2.2), the Li and Mak portmanteau test and our LM test collapse into the

McLeod and Li (1983) and the Engle test, respectively.

Our test or the Li and Mak test are in principle general misspecication tests but,

as is clear from above, they also have an LM interpretation, that is, they are LM tests

against a certain parametric alternative. If the model builder testing his or her GARCH

model believes that the only relevant alternative is a higher order GARCH model,

then the LM test of Bollerslev (1986) is an obvious test to use. If a more general

misspecication of the model is not excluded a priori, then the test of this section can

be viewed as a useful complement to Bollerslevs test. Other types of misspecication

will be considered next.

4. Misspecication of structure

In this section, we present two misspecication tests for an estimated conditional

variance model. The rst one is a test against nonlinearity or, in some cases, asym-

metry. It is a modication of a test in Hagerud (1997). Second, we propose a test of

parameter constancy against smooth continuous change in parameters. These tests may

be viewed as conditional variance counterparts of the tests for the nonlinear conditionalmean presented in Eitrheim and Terasvirta (1996). To describe the common features

in these tests we rst introduce some notation and thereafter consider the two tests

separately.

4.1. Notation

Consider now an augmented version of model (2.2),

t = zt(ht + gt)1=2; (4.1)

where gt = g(?1?2; W; st) such that g(0; ?2; W; st) 0, and st is a vector of randomvariables. The null hypothesis to be tested, using this notation, is ?1 =0. As an example,

the alternative in the LM test of Bollerslev (1986) for testing a GARCH( p; q) model

against GARCH(p; q + r) or GARCH(p + r; q); r 0, alternatives, belongs to class

(4.1). As another example, one could think of a Ramsey-type functional form LM

test, in which case st = st, and gt = 1(Wst)

2 or gt = (Wst)[exp{1(Wst)} 1], where

1 is a scalar and 2 = 0. The null hypothesis would, as above, be 1 = 0. This would

lead to a one-degree-of-freedom test. When applying GARCH models the degrees of

freedom are not usually a problem, however, so that we may consider more richly parameterized alternatives. We shall concentrate on tests of linearity and parameter

constancy of the GARCH(p; q) model.


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4.2. Testing linearity

When volatility in return series is modelled with GARCH models, we may sometimes

expect the response to be a function not only of the size of the shock but also of thedirection. Engle and Ng (1993), see also references therein, considered this possibility.

We call such a response to a shock asymmetric and parameterize it by generalizing

the GJRGARCH model of Glosten et al. (1993). This is done in three ways. First,

we make the transition between the extreme regimes smooth. Second, we incorporate a

nonlinear version of the quadratic GARCH model of Sentana (1995) in our alternative.

Finally, while the GJRGARCH model is asymmetric, the present generalization may

in some special cases remain symmetric, although the model is nonlinear. For smooth

transition GARCH, see also Hagerud (1997), Gonzalez-Rivera (1998) and Anderson et

al. (1999). Our maintained model is in fact a special case of the model of Anderson

et al. (1999). Let

Hn(xt; ; c) =

1 + exp

nl=1

(xt cl)

1

; 0; c16 6 cn; (4.2)

where xt is the transition variable at time t, is a slope parameter, and c = (c1; : : : ; cn)

a location vector. Conditions 0 and c16 6 cn are identifying conditions. When = 0; Hn(xt; ; c) 12 . Typically in practice, n =1 or n =2. The former choice yields astandard logistic function. When the slope parameter , (4.2) with n =1 becomesa step function whose value equals one for xt

c1 and zero otherwise. When n = 2,(4.2) is symmetric about (c1 + c2)=2, its minimum value, achieved at this point, lies

between zero and 12

, and the value of the function approaches unity as xt . When , function (4.2) becomes a double step function that obtains value zero forc16xt c2 and unity otherwise. The logistic function (4.2) is used for parameterizing

the maintained model. Let Hn = Hn 12 . This transformation simplies notation inderiving the test but does not aect the generality of the argument. The alternative to

standard GARCH may now be written as (4.1) where

gt =

q

j=1

0j Hn(tj; ; c) +

q

j=1

1j Hn(tj; ; c)2tj : (4.3)

Using previous notation, ?1 = and ?2 = (c1; : : : ; cn; 01; : : : ; 0q; 11; : : : ; 1q) in gt,

and st=st. The sucient but not necessary conditions for ht+gt 0 are 0 0;q

i=1 0i 0; j 0; j + 1j 0; j = 1; : : : ; q. Assuming n = 1; 0j = 0; j = 1; : : : ; q, and letting

in (4.3) yields the GJRGARCH model. The test of the standard GARCHmodel against nonlinear GARCH in Hagerud (1997) may be viewed as a special case

of this specication with gt =q

j=1 1jHn(tj; ; c)

2tj.

Another way of parameterizing the alternative is to assume that the transition variable

has a xed delay. This assumption results in the following conditional variance model:

gt = 0d Hn(td; ; c) +q

j=1

1j Hn(td; ; c)2tj : (4.4)


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This specication resembles the one in Terasvirta (1994) for the STAR-type conditional

mean. Probably the most common case in practice, however, is d = q = p = 1 so that

(4.3) and (4.4) coincide. Note that we do not impose any nonlinear structure on the

htj ; j = 1; : : : ; p, as the alternative model is already very exible without such anextension.

These smooth transition alternatives pose an identication problem. The null hypo-

thesis can be expressed as H0: = 0 in Hn. It is seen that the remaining parameters

in (4.2) or (4.3) are only identied under the alternative. In other words, the ele-

ments of ?2 in gt are nuisance parameters that do not appear in the null model. Thus,

the classical test statistics are not available as their asymptotic null distributions are

unknown; see, for example, Hansen (1996) for discussion. We circumvent the identi-

cation problem by following Luukkonen et al. (1988a). This is done by expanding

the transition function Hn

into a rst-order Taylor series around = 0, replacing the

transition function with this Taylor approximation in (4.3) and rearranging terms. As

a result,

ht = (W)st and gt = R

1v1t +

n+2i=3

Ri vit + R1 (4.5)

in (4.1) and, furthermore, Ri = (i1; : : : ; iq) = Ri; Ri = 0; vit = (it1; : : : ; itq); i =

1; 3; : : : ; n + 2, and R1 is the remainder. The new null hypothesis, implied by (4.5),

is H0: R1 = R3 = = Rn+2 = 0. Note that under H0:R1 = 0, so that the stochasticremainder does not aect the distributional properties of the test statistic when thenull hypothesis holds. With ht and gt dened as in (4.5), Eq. (4.1) may be called an

auxiliary GARCH equation. We make the following additional assumption:

(A.1) Under H0; E2(n+2)t .

Remark. The restrictions Assumption (A.1) implies on the parameters in the standard

GARCH(1; 1) model appeared in Nelson (1990); see also He and Terasvirta (1999).

Recently; Carrasco and Chen (2002) worked out conditions for the existence of mo-

ments of higher order than four for the GARCH(p; q) model.

We dene the conditional quasilog-likelihood for observation t as follows:

t = (1=2)ln(ht + gt) 2t

2(ht + gt): (4.6)

An LM test for the null hypothesis may now be derived starting from (4.6) along the

same lines as in Appendix A. Note that gt is not a function of W and that under

H0; gt 0. Here, xt = h1t 9ht=9W (9ht=9W denotes 9(ht + gt)=9W evaluated underH0). In fact, xt has exactly the same form as in Theorem 3.1 because the null model

is the GARCH(p; q) model in both cases. On the other hand, vt = (v

1t; v

3t; : : : ; v

n+2; t).

Finally, Assumption (A.1) is needed for the existence of Evn+2; tv

n+2; t.

Theorem 4.1. Consider the auxiliary GARCH equation (4.1) where {zt} is a sequenceof independent identically distributed random variables with zero mean; unit variance


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and Ez3t = 0; with ht and gt are dened as in (4.5). Under the assumption (A.1) and

H0: R1 = R3 = = Rn+2 = 0 in (4.5); the statistic (3.3) with xt as in Theorem 3.1 andvt = (v

1t; v

3t; : : : ; v

n+2; t) with vit = (

it1; : : : ;

itq)

; i = 1; 3; : : : ; n + 2; has an asymptotic

2 distribution with m = (n + 1)q degrees of freedom.

A similar result may be derived for the xed delay case. It is also possible to test

subhypotheses. For example, if the alternative to GARCH is the modied quadratic

GARCH model in which only the intercept displays a smooth transition, the null hy-

pothesis equals R1 = 0, whereas R3 = = Rn+2 = 0 even in the maintained model.This test is a linearity test whose alternative generally implies asymmetry. Note,

however, that if n = 2 and c1 = c2 in (4.2), the smooth transition GARCH modelis still symmetric. Such a model may be useful, for example, in situations where the

large shocks die out so much more quickly than small shocks that the response cannot be adequately characterized by linear GARCH models.

The test may be carried out in the TR2 form via an auxiliary regression exactly

as in the previous section. The same is true for the robust version of the linearity

test.

4.3. Testing parameter constancy

Testing parameter constancy is important in its own right but also because noncon-

stancy may manifest itself as an apparent lack of covariance stationarity (IGARCH);

see, for example, Lamoureux and Lastrapes (1990). Here, we assume that the alter-native to constant parameters in the conditional variance is that the parameters, or a

subset of them, change smoothly over time. Lin and Terasvirta (1994) applied this idea

to testing parameter constancy in the conditional mean. Let the time-varying parameter

W(t) =W+[ Hn(t; ; c). If the null hypothesis only concerns a subset of parameters then

only the corresponding elements in [ are assumed to be nonzero a priori. Again, we

dene Hn =Hn 12 where the transition function Hn(t; ; c) is assumed to be a logisticfunction of order n dened in (4.2) with xt t. When ; H1(t; ; c) becomes astep-function and characterizes a single structural break in the model. Chu (1995) and

Lin and Yang (1999) discussed testing parameter constancy against this alternative.

The null hypothesis of parameter constancy becomes H0: = 0 under which W(t) W.By dening ?1 = and ?2 = ([

; c) we can consider this as a special case of (4.1)

with gt = ([st) Hn(t; ; c). We can again circumvent the lack of identication under

the null hypothesis by a Taylor approximation of the transition function. A rst-order

Taylor-expansion of Hn(t; ; c) around = 0 yields, after a reparameterization, model

(4.1) with

ht = (W)st and gt =

ni=1

Ri vit + R2: (4.7)

In (4.7), Ri = Ri; Ri = 0; i = 1; : : : ; n. The null hypothesis based on (4.7) equals R1 = = Rn = 0, and the remainder R2 = 0 when the null hypothesis is valid. Furthermore,vit = t

ist; i = 1; : : : ; n. Dene vit = ti st; i = 1; : : : ; n, and vt = (v

1t; : : : ; v

nt).


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The conditional quasilog-likelihood has the form (4.6), and we can now derive an

LM test for parameter constancy using the denitions in (4.7). This yields the following

result:

Theorem 4.2. Consider the auxiliary GARCH equation (4.1) with (4.7) where {zt}is a sequence of independent identically distributed random variables with zero mean;

unit variance andEz3t = 0. Under assumption (A.2) and H0: R1 = = Rn = 0 in (4.7);the statistic (3.3) with xt as in Theorem 3.1 and vt=(v1t; : : : ; vnt)

; vit=ti st; i =1; : : : ; n ;

has an asymptotic 2 distribution with m = n(p + q + 1) degrees-of-freedom.

Remark 1. The proof requires the assumption Ests

t ; which is equivalent to assum-ing E4t . Since the regularity conditions assumed throughout require even higherorder moments; we do not explicitly mention this condition in the theorem.

Remark 2. A number of terms in the auxiliary GARCH equation now contain trending

variables. Nevertheless; applying the results of Lai and Wei (1982) as when proving a

corresponding result for linear conditional mean models; see Lin and Terasvirta (1994);

it can be shown that the asymptotic null distribution even in this case is a chi-squared

one. This makes the test easy to apply.

Even this test may be carried out using an auxiliary regression as in Section 3.1.

The procedure for carrying out the robust test, described in Section 3.1, is also valid

for the parameter constancy test.An advantage of a parametric alternative to parameter constancy is that if the null

hypothesis is rejected we can estimate the parameters of the alternative model. This

helps us nd out where in the sample the parameters under test have been changing and

how rapid the change appears to be. This is useful information when respecication of

the model to achieve parameter constancy is attempted.

5. Simulation experiment

The above distribution theory is asymptotic, and we have to nd out how our tests behave in nite samples. This is done by simulation. For all simulations we use the

following data generating process (DGP)

yt = t;

where t follows a standard GARCH process (size simulations) or one of the alter-

natives discussed above (power). The random numbers are generated by the random

number generator in GAUSS 3.2.31. The rst 200 observations of each generated series

are always discarded to avoid initialization eects. All experiments are performed with

series of 1000 observations. For each design, a total of 1000 replications have been

carried out. The distribution for the error process {zt} is either standard normal or astandardized (unit variance) t(d) distribution, where d = 3; 4; 5. We report results from

experiments with normal and t(3)-errors. The ones with the other two t-distributions


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lie between these two extremes. Note that for t(3) and t(4)-errors, the fourth moment

of t does not exist, which contradicts our assumptions. The idea is to investigate the

eect of this violation of the conditions on the results.

Both the normality-based LM and the robust version of each test are considered.This is also true for the tests to which we compare our tests. A general result valid for

practically all simulation experiments is that the nonrobust test is undersized when the

error distribution is any one of the three t-distributions. Note that the results appearing

in the gures are based on size-adjusted tests, so that this nding is not visible there.

The robust tests are more powerful than the original LM or LM type tests. A general

conclusion from our experiments is that in applied work one should apply the robust

versions of our tests.

5.1. No remaining ARCH

First, we consider the test of no ARCH in the standardized errors. We dene a DGP

such that the conditional variance follows a GARCH(1; 2) process:

ht = 0:5 + 0:052t1 +

2t2 + 0:9ht1: (5.1)

In the Monte Carlo experiment the values of vary within limits such that the condi-

tional variance of the process remains positive with probability 1, see Nelson and Cao

(1992), and covariance stationarity holds. This is the case when 0:045 0:05.For = 0 the DGP reduces to a standard GARCH(1; 1) model. The results for the test

of no ARCH in the standardized errors are reported in Fig. 1. The nominal signicance

level equals 0.1 (the results for levels 0.05 and 0.01 are not reported), and we use

the test of Bollerslev (1986) against the GARCH(1; 2) model as a benchmark. In the

simulations, it was always assumed that the additional ARCH component in our test

was ARCH(1), so that vt= 2t1=ht1. Robust and nonrobust versions of both tests were

simulated.

Our results indicate that the robust tests are well-sized, whereas the nonrobust ones

are undersized when the error distribution is a t-distribution. The size-adjusted power

results appearing in Fig. 1 are somewhat ambiguous in the normal case. The nonrobust

tests are more powerful than the robust ones for negative values of but less powerfulfor positive values of this parameter. This is true for both Bollerslevs tests and ours.

When the errors are t-distributed, a clear dierence in power in favour of the robust

tests emerges. Note, however, that in that case both tests appear somewhat biased. The

two tests in our comparison seem to have very similar size-adjusted power, although

our test has somewhat greater power when the errors follow a t(3)-distribution. As

Bollerslevs test is an LM test against the data-generating alternative, we may conclude

that our test works very well in this experiment.

5.2. Testing linearity

Our linearity test can be expected to be powerful against smooth transition alterna-

tives for which it is designed. Assessing its performance against, say, the GJRGARCH


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Fig. 1. Results from size and power simulations of the LM test of no remaining ARCH (circle) and the

LM test of GARCH(1,1) against GARCH(1,2) (cross). DGP: GARCH(1,2), Eq. (5.1). x-axis: , y-axis:

size-adjusted power. Upper panel, left: normal errors, nonrobust test, upper panel, right: normal errors,

robust test, lower panel, left: t(3)-errors, nonrobust test, lower panel, right: t(3)-errors, robust test. Number

of observations=1000, number of replications=1000.

model would constitute a tougher trial for the test. This would also make it possible

to compare the performance of the tests with the sign-bias and negative size-bias tests

of Engle and Ng (1993). The DGP is, accordingly, (2.2) withht = 0:005 + 0:23[|t1| + !t1]2 + 0:7ht1; (5.2)

where t is assumed conditionally normal. In Eq. (5.2), |!| 0:551 is required forcovariance stationarity. This is the model Engle and Ng (1993) used in their simulation

except that the coecient of [|t1| + !t1]2 equals 0.23 against 0.28 in Engle andNg (1993). This change guarantees the existence of the unconditional fourth moment

under the null hypothesis ! = 0; the relevant moment condition appeared in He and

Terasvirta (1999). (The sign-bias and size-bias tests require a nite fourth moment.)

The sign-bias and size-bias tests are designed for detecting asymmetry in the conditional

variance. We compute the values of these test statistics using the quadratic form. Thedierence in results between this test and the TR2 version, suggested by Engle and Ng

(1993), is negligible at our sample size. Engle and Ng (1993) used (2.2) and (5.2)


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Fig. 2. Results from size and power simulations of the LM-type parameter constancy test (circle) and the

robust version of the parameter constancy test of Chu (1995) (cross). DGP: GARCH(1,1) with a single

structural break, equation (5.3). x-axis: , y-axis: size-adjusted power. Upper panel, left: normal errors,

nonrobust test, upper panel, right: normal errors, robust test, lower panel, left: t(3)-errors, nonrobust test,

lower panel, right: t(3)-errors, robust test. Number of observations=1000, number of replications=1000.

with !=

0:23 as the DGP in their evaluation of the sign-bias test. We also considered

robust versions of the two tests.The results for the sign-bias and negative size-bias tests are rather similar. Thus, the

latter test is excluded from Fig. 2 where our test is compared with the sign-bias test.

The nonrobust sign-bias and our linearity test are undersized for the t(3)-errors. The

robust versions of the tests have roughly the correct size.

As for size-adjusted power, results for ! 0 can be found in Fig. 2. For positive

values of ! the results are similar and therefore omitted. Our smooth transition GARCH

test was computed by assuming n = 1 in (4.2). When the errors are normal, 1000

observations are enough for the robust tests to perform as well as the nonrobust ones.

In the case of t-distributed errors, the robust tests are always more powerful than

the nonrobust ones. Our linearity test is clearly more powerful than the sign-bias testeven if the alternative is a GJRGARCH and not a genuine smooth transition GARCH

model.


14/19


Fig. 3. Results from power simulations of the LM-type parameter constancy test (circle) and the parameter

constancy test of Chu (1995) (cross). DGP: GARCH(1,1) with a double structural break, Eq. (5.3). x-axis:

, y-axis: size-adjusted power. Upper panel, left: normal errors, nonrobust test, upper panel, right: normal

errors, robust test, lower panel, left: t(3)-errors, nonrobust test, lower panel, right: t(3)-errors, robust test.

Number of observations=1000, number of replications=1000.

5.3. Testing parameter constancy

We consider two cases of parameter nonconstancy: the DGP is a GARCH(1; 1)

model with either (a) a single or (b) a double structural break. We did not simulate

smooth parameter change because our test can be expected to perform well against

such an alternative. Our choice of alternative gives us an opportunity to compare our

test with tests against a single structural break, and we choose the test of Chu (1995)

for the purpose. If T is the total number of observations, the single structural break

parameterization (a) is assumed to have a break at time T where lies between 0

to 1. The double structural break parameterization (b) postulates a change at time 1T

and a return to the original parameters at 2T; 06 1 26 1. Our test is computedwith n = 1, case (a), and n = 2, case (b), where n is the order of the logistic function

in (4.2).


15/19


Fig. 4. Results from size and power simulations of the LM-type linearity test (circle) and the sign-bias test

of Engle and Ng (1993) (cross). DGP: GJRGARCH(1,1), Eq. (5.2). x-axis: !, y-axis: size-adjusted power.

Upper panel, left: normal errors, nonrobust test, upper panel, right: normal errors, robust test, lower panel,

left: t(3)-errors, nonrobust test, lower panel, right: t(3)-errors, robust test. Number of observations=1000,

number of replications=1000.

We consider the following model for a change in the constant term:

ht = 0:5 + 0:12t1 + 0:8ht1; (a) t T; (b) t 1T; t 2T;

ht = 0:5(1 + ) + 0:12t1 + 0:8ht1; (a) t T; (b) 1T6 t6 2T; (5.3)

where = 0:4; 0:8. Chu (1995) used (a) in (5.3) as the DGP in his own simulation

experiments. The power simulations for the DGPs in (5.3) with a single structural

break at for = 0:8 appear in Fig. 3. The values = 0; 1 correspond to the null

hypothesis. Even if only the intercept changes in (5.3), in simulating our LM-type

test we assume that under the alternative, the break aects all three parameters. The

asymptotic null distribution of our LM-type statistic is thus the 2(3)-distribution.

We report results for = 0:8. The pattern of the results for = 0:4 is similar, only

the power is lower. In another experiment, we allowed the coecient of 2t1 to changeonce within the sample period. The behaviour of the tests was similar to the previous

case, and no details are given here. In case (a) under normality, our LM-type test


16/19


has the same as or higher power than the test of Chu (1995) for 0:36 6 0:7, but

otherwise the relationship is the opposite. When the errors follow a t(3)-distribution,

the robust version of our test is more powerful than the standard version. It is also again

more powerful than Chus test for 0:36 6 0:7. It seems, however, that Chus robusttest is oversized when the errors are t(3)-distributed whereas our robust one is not.

On the other hand, in that situation the nonrobust LM-type test is clearly undersized

(these two facts cannot be seen from Fig. 3).

We turn to case (b), the double structural break. The DGP in that design is such

that 2 = 1 + 0:3 where 1 varies from 0 to 0:7. Thus, for 1 =0 and 1 = 0:7, the DGP

only has a single structural break. Results of power simulations of this design with

= 0:8 can be found in Fig. 4. In the case of a double break the test of Chu (1995)

cannot be expected to be very powerful because the test is constructed for detecting

just a single structural break. For this reason, its power is higher for 1 close to zero

and 0:7 than elsewhere. With few exceptions (nonrobust test, t(3)-distributed errors)

our test with n =2 does have size-adjusted power superior to that of Chu for all double

break points considered. For t-distributed errors, the robust version of the test is clearly

more powerful than the nonrobust one.

6. Conclusions

In this paper, we derive a number of misspecication tests for a standard GARCH

(p; q) model. As all our test statistics are asymptotically 2-distributed under the null

hypothesis, possible misspecication of a GARCH model can be detected at low com- putational cost. Because the tests of linearity and parameter constancy are parametric,

the alternative may be estimated if the null hypothesis is rejected. This is useful for

a model builder who wants to nd out the possible weaknesses of the estimated spec-

ication. It may also give him=her useful ideas of how the model could be further

improved.

We also highlight the fact that our test of no ARCH in the standardized error process

is asymptotically equivalent to a portmanteau test of Li and Mak (1994). This links

the work of these authors to our approach. As already mentioned, the advantage of our

derivation of the no ARCH test statistic is that robustifying the test against nonnormal

errors is straightforward.

Finally, the simulation results indicate that in practice, the robust versions of the tests

should be preferred to nonrobust ones. At relevant sample sizes when the errors are

normal, they are roughly as powerful as normality-based LM or LM type tests. When

the errors are nonnormal, the robust tests are superior in terms of power to nonrobust

ones. They can therefore be recommended as standard procedures when it comes to

testing the adequacy of an estimated standard GARCH(p; q) model.

Acknowledgements

The rst author acknowledges nancial support from the Tore Browaldhs Foundation

and Skandia Asset Management. The research of the second author has been supported


17/19


by the Swedish Council for Research in Humanities and Social Sciences. We thank

James Davidson and an anonymous referee for helpful suggestions and remarks but

retain the responsibility for any errors and shortcomings in the paper.

Appendix A. Proof of Theorem 3.1

The conditional quasi log likelihood for observation t equals

t = 12

ln ht 12

ln gt 2t

2htgt:

The partial derivative with respect to W is

9t

9W= 1

2

1

ht

9ht

9W+

1

gt

9gt

9W

2t

h2tgt

1

ht

9ht

9W

2t

htg2t

1

gt

9gt

9W

=1

2(2t =htgt 1)xt +

1

2(2t=htgt 1)

1

gt

9gt

9W;

where xt = h1t 9ht=9W with 9ht=9W = st + (9st=9Wt)

W, and 9gt=9W = (9vt=9Wt)?. Thus,

9gt=9W|H0 = 0 so that 9t=9W|H0 = ( 12 )(2t=ht 1)xt. Likewise,

9t9?

= 12

(2t=htgt 1) 1gt 9gt

9?

so that 9t=9?|H0 = ( 12 )(2t=ht 1)vt, where vt = (2t1=ht1; : : : ; 2tm=htm). Let G =(D; W). The relevant component of the average pseudoscore evaluated at the true

(under the null hypothesis) pair of parameter values (G0; 0) has the form

qT(G0; 0) =1

2T

Tt=1

(2t=ht 1)

x0t

vt

;

where x0t equals xt evaluated at (G0; 0). (We need not consider (1=T)

Tt=1 9t=9D|H0 ,

because assuming Ez3t = 0 makes the covariance matrix of the full pseudoscore vector

block diagonal.) Let t = yt f(wt; D) where D is the quasi maximum likelihoodestimator ofD under H0. Furthermore, let ht; xt and vt equal ht; xt and vt evaluated at

G = G. Under regularity conditions,

TqT(G; 0) =

1

2

T

Tt=1

(2t=ht 1)

xtvt

=

0

(1=2

T)

Tt=1

(2t=ht 1)vt N(0; J0);


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in distribution as T , where

J0 = J011 J012J021 J022

= ET1qT(G0; 0)qT(G0; 0)

=k

4E

x0t x

0

t x0t v

t

x0tv

t vtv

t

:

with k = E(2t=ht 1)2. Thus,

LM0 =

T

t=1

(2t=ht 1)vt)J1022

T

t=1

(2t=ht 1

vt

(A.1)

has an asymptotic 2

distribution with m degrees of freedom. Replacing J1

022 in (A.1) by a consistent estimator

V(W)1 = (4T=k)

Tt=1

vtv

t T

t=1

vtx

t

T

t=1

xtx

t

1 Tt=1

xtv

t

1

with k= (1=T)T

t=1(2t=ht1)2, yields the result; see Davidson (2000, Section 12.3.3).

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